circumnavigation from distance measurements under slow drift

Circumnavigation From Circumnavigation From Distance Measurements Distance Measurements Under Slow DriftUnder Slow Drift

Soura Dasgupta, U of IowaSoura Dasgupta, U of IowaWith: Iman Shames, Baris Fidan, Brian With: Iman Shames, Baris Fidan, Brian

AndersonAnderson

Outline• The Problem

– Motivation– Precise Formulation

• Broad Approach• Localization• Control Law• Analysis

– Stationary target– Drifting target

• Rotation selection• Simulation• Conclusion

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Problem

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Problem

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Problem

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Problem

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Problem

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Problem

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Problem

Sufficiently rich orbitSufficiently rich perspective

Slow unknown drift in target

2 and 3 dimensionsANU July 31, 2009 9 of 27

Motivation• Surveillance• Monitoring from a distance• Require a rich enough perspective• May only be able to measure distance

– Target emitting EM signal– Agent can measure its intensity Distance

• Past work– Position measurements– Local results– Circumnavigation not dealt with

• Potential drift complicates

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If target stationary Measure distances from three noncollinear agent positions

In 3d 4 non-coplanar positions

Localizes target

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If target stationary

Move towards target

Suppose target drifts

Then moving toward phantom position

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Coping With Drift• Target position must be continuously estimated

• Agent must execute sufficiently rich trajectory– Noncollinear enough: 2d– Noncoplanar enough: 3d

• Compatible with goal of circumnavigation for rich perspective

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Precise formulation• Agent at location y(t)• Measures D(t)=||x(t)-y(t)||• Must rotate at a distance d from target• On a sufficiently rich orbit• When target drifts sufficiently slowly

– Retain richness– Distance error proportional to drift velocity

• Permit unbounded but slow drift

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Quantifying Richness

• Persistent Excitation (p.e.)• The i are the p.e. parameters • Derivative of y(t) persistently spanning• y(t) avoids the same line (plane) persistently• Provides richness of perspective• Aids estimation

IdyyITt

t21 )(')(0

1

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Outline• The Problem

– Motivation– Precise Formulation

• Broad Approach• Localization• Control Law• Analysis

– Stationary target– Drifting target

• Rotation selection• Simulation• Conclusion

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Broad approach• Stationary target

• From D(t) and y(t) localize agent

• Force y(t) to circumnavigate as if it were x

xtx )(ˆ

)(ˆ tx

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Coping With drifting Target• Suppose exponential convergence in stationary

case• Show objective approximately met when target

velocity is small

• x(t) can be unbounded• Inverse Lyapunov arguments• Wish to avoid partial stability arguments

|)(|)()( txKdtxty

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Outline• The Problem

– Motivation– Precise Formulation

• Broad Approach• Localization• Control Law• Analysis

– Stationary target– Drifting target

• Rotation selection• Simulation• Conclusion

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Rules on PE• R(t) p.e. and f(t) in L2 R(t)+f(t) p.e.

– L2 rule

• R(t) p.e. and f(t) small enough R(t)+f(t) p.e.– Small perturbation rule

• R(t) p.e. and H(s) stable minimum phase H(s){R(t)} p.e.– Filtering rule

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A basic principle• Suppose x(t) is stationary and• We can generate

• Then:

))(ˆ)(()( xtxtztv T

)()()(ˆ tvtztx

xtx )(ˆ If z(t) p.e.

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Localization• Dandach et. al. (2008)• If x(t) stationary• Algorithm below converges under p.e.• Need explicit differentiation

))()(()()(21

))(()')(()(2

xtytytDtD

xtyxtytD

T

))(ˆ)(())(ˆ)()(()()(21 xtxtytxtytytDtD TT

)))(ˆ)()(()()(21)(()(ˆ txtytytDtDtytx T

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Localization without differentiation

)(21)()()(

||)(||21)()()(

)(21)()()(

33

222

211

tytztztV

tytztztm

tDtztzt

))(ˆ)((

)(ˆ)()()(

xtxtV

txtVtmtT

T

))(ˆ)()()()(()(ˆ txtVtmttVtx T

xtx )(ˆ If V(t) p.e. p.e. )(ty

x stationary

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Summary of localization• Achieved through signals generated

– From D(t) and y(t)– No explicit differentiation

• Exponential convergence when derivative of y(t) p.e.– x stationary– Implies p.e. of V(t)

• Exponential convergence robustness to time variations – As long as derivative of y is p.e.

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Outline• The Problem

– Motivation– Precise Formulation

• Broad Approach• Localization• Control Law• Analysis

– Stationary target– Drifting target

• Rotation selection• Simulation• Conclusion

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Control Law• How to move y(t)?• Achieve circumnavigation objective around• A(t)

– skew symmetric for all t– A(t+T)=A(t)– Forces derivative of z(t) to be p.e.

)(ˆ tx

)(ˆ)( txty ))(ˆ)()()(ˆ( 22 txtydtD ))(ˆ)()(( txtytA

||)(ˆ)(||)(ˆ txtytD

)()()( tztAtz ANU July 31, 2009 28 of 27

The role of A(t)

• A(t) skew symmetric

• Φ(t,t0) Orthogonal

• ||z(t)||=||z(t0) ||

• z(t) rotates

)(),()(

)()()(

00 tztttz

tztAtz

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Control Law Features• Will force

• Forces Rotation• Overall still have

• p.e. • Regardless of whether x drifts

)(ˆ)( txty ))(ˆ)()()(ˆ( 22 txtydtD ))(ˆ)()(( txtytA

)()()( tztAtz

dtD )(ˆ

dtD )(ˆ

)(ˆ)( txty

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Closed Loop

)(21)()()(

||)(||21)()()(

)(21)()()(

33

222

211

tytztztV

tytztztm

tDtztzt

))(ˆ)()()()(()(ˆ txtVtmttVtx T

))(ˆ)()(())(ˆ)()()(ˆ()(ˆ)( 22 txtytAtxtydtDtxty

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Nonlinear Periodic

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Outline• The Problem

– Motivation– Precise Formulation

• Broad Approach• Localization• Control Law• Analysis

– Stationary target– Drifting target

• Rotation selection• Simulation• Conclusion

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The State Space

)(21)()()(

||)(||21)()()(

)(21)()()(

33

222

211

tytztztV

tytztztm

tDtztzt

))(ˆ)()()()(()(ˆ txtVtmttVtx T

))(ˆ)()(())(ˆ)()()(ˆ()(ˆ)( 22 txtytAtxtydtDtxty

)(

)(ˆ)(~)()()(

3

2

1

tyxtxtx

tztztz

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Looking ahead to drift• When x is constant• Part of the state converges exponentially to a point• Part (y(t)) goes to an orbit• Partially known

– Distance from x– P.E. derivative

• Standard inverse Lyapunov Theory inadequate• Partial Stability?• Reformulate the state space

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Regardless of drift

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dtxty ||)(ˆ)(||

)(ˆ)( txty p.e.

y(t) circumnavigates )(ˆ tx

Stationary case: Need to showDrifting case: Need to show

xtx )(ˆ

small is )( toclose gets )(ˆ

txxtx

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Globally

Stationary Analysis

• p(t)=η(t)-m(t)+VT(t)x(t) V(t) p.e.

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2ˆ

)()(~

0)()()(

)()(~

Lx

tptxtVtVtV

tptx T

0)()(~

tptx

)(ˆ)( txty p.e. p.e.)(ty

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Nonstationary Case• Under slow drift need to show that derivative of

y(t) remains p.e • Tough to show using inverse Lyapunov or partial

stability approach• Alternative approach: Formulate reduced state

space– If state vector converges exponentially then objective

met exponentially– If state vector small then objective met to within a small

error• y(t) appears as a time varying parameter with

proven characteristicsANU July 31, 2009 37 of 27

Key device to handle drift

• q(t) p.e. under small drift• Reformulate state space by replacing derivative of y(t) by

• q(t) is p.e. under slow enough target velocity• Partial characterization of “slow enough drift”

– Determined solely by A(t), and d

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)()(ˆ)()(

)()(ˆ)()(

txtxtqty

txtxtytq

)()(ˆ)()(1 txtxtqtq

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Reduced State Space• q(t) p.e. under small drift• r(t)=1/(s+α){q(t)} p.e.• Reduced state vector:

• Stationary dynamics:– eas when r(t) p.e.

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)(~)()(

],~,[

txtwtw

pxw TT

)())(),(()( tttrFt

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Reduced State Space• q(t) p.e. under small drift• r(t)=1/(s+α){q(t)} p.e.• Reduced state vector:

• Nonstationary dynamics• G and H linear in • Meet objective for slow enough drift

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)(~)()(

],~,[

txtwtw

pxw TT

)()()]())(),(([)( xHtxGttrFt

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x

Outline• The Problem

– Motivation– Precise Formulation

• Broad Approach• Localization• Control Law• Analysis

– Stationary target– Drifting target

• Rotation selection – Selecting A(t)

• Simulation• Conclusion

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Selecting A(t)

• A(t):– Skew symmetric– Periodic– Derivative of z p.e.– P.E. parameters depend on d

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)(ˆ)( txty ))(ˆ)()(( txtytA

)()()( tztAtz 2

22

1 )0()()()0(01

zdsszszz TTt

t

))(ˆ)()()(ˆ( 22 txtydtD

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2-Dimension

• A(t):– Skew symmetric– Periodic– Derivative of z p.e.

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)()()( tztAtz

0110

)( ctA

Tctcttztz )sin()cos()()( 0

Constant

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3-Dimension

• A(t):– Skew symmetric– Periodic– Derivative of z p.e.

• Will constant A do?– No!– A singular Φ(t) has eigenvalue at 1

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)()()( tztAtz

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A(t) in 3-D• Switch periodically between A1 and A2

• Differentiable switch • To preclude impulsive force on y(t)

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000001010

11 aA

010100000

22 aA

))(ˆ)()(())(ˆ)()()(ˆ()(ˆ)( 22 txtytAtxtydtDtxty

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Outline• The Problem

– Motivation– Precise Formulation

• Broad Approach• Localization• Control Law• Analysis

– Stationary target– Drifting target

• Rotation selection• Simulation• Conclusion

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Circumnavigation Via Distance Measurements

Distance Measurements

Target Position Error

Trajectories

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Circumnavigation Via Distance Measurements

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Circumnavigation Via Distance Measurements

Distance Measurements

Target Position Error

Trajectories

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Circumnavigation Via Distance Measurements

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The Knee

• Initially this dominates– Zooms rapidly toward estimated location

• Fairly quickly rotation dominates

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)(ˆ)( txty ))(ˆ)()()(ˆ( 22 txtydtD ))(ˆ)()(( txtytA

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Conclusions• Circumnavigation• Distance measurements only• Rich Orbit• Slow but potentially unbounded drift• Future work

– Designing fancier orbits– Positioning at a distance from multiple objects– Noise analysis

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